Wrapping flat artwork onto a sphere with no distortion to create a football
Yes, you can model and texture a football in Blender with any shape you like.
The 32 panel standard ball is probably the easiest to accomplish, since there's a standard primitive (the Icosphere) that pretty much resembles it's shape.
Here is a tutorial explaining how to do it https://www.youtube.com/watch?v=AYC69tt-KMI

Other shapes can also be accomplished too, but it's process can be more or less involved depending on the complexity of the shape.

About texturing, as mentioned by Mr Zak what you need is probably UV Unwrapping, this can also be accomplished inside Blender. It is the process of "unfolding" or "flattening" an 3D geometry into a 2D shape so that an image texture can be mapped over it's surface as desired with minimal distortion.

Depending on how big your design is, or how much of the ball it covers unwrapping could be a very simple process. For a simple logo that sits only on a side of a ball a simple view-based direct projection could be enough. For designs that cover the whole ball it could be a more complex process.

Either way you could probably model a few different predefined football geometries with the most commonly used shapes, and unwrap them in the most convenient way to simplify the surface design. You could then use that UV layout as a template save it as an image or even export as vector using the SVG export addon (if you work on vector graphics design software) and reuse it whenever you need to design a ball for a specific geometry.
All you need to do then is import that layout into your graphic design application (Inkscape, GIMP, Photoshop, Illustrator, whatever) and draw over it so the parts of the design match the desired corresponding area in the ball surface.

The base 3D models can also be reused as asset libraries to easily render your designs whenever you need. You can even have a set of predefined files with your scene (grass, field, product studio, etc), material, and lighting setup all ready to render, and all you need to do is apply your new surface image texture to it and render your design. You only need to set it up once.